Noncompact Lie Groups and Some of Their Applications

Noncompact Lie Groups and Some of Their Applications

NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences B Physics C Mathematical and Physical Sciences D Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences G Ecological Sciences H Cell Biology I Global Environmental Change Plenum Publishing Corporation London and New York Kluwer Academic Publishers Dordrecht, Boston and London Springer-Verlag Berlin, Heidelberg, New York, London, Paris and Tokyo NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: - via online FILE 128 (NATO-PCO-DATA BASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. - via CD-ROM "NATO-PCO-DATA BASE" with user-friendly retrieval software in English, French and German ( WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series C: Mathematical and Physical Sciences - Vol. 429

Noncompact Lie Groups and Some of Their Applications edited by Elizabeth A. Tanner and Raj Wilson Division of Mathematics, University of Texas, San Antonio, Texas, U.SA Springer Science+Business Media, B.v.

Proceedings of tha NATO Advanced Resaarch Workshop on Noncompact Lie Groups and Their Physical Applications San Antonio, Texas, U.SA January 4-8, 1993 A C.I.P. Catalogue record for this book is available from the Ubrary of Congress. ISBN 978-94-010-4470-7 DOI 10.1007/978-94-011-1078-5 ISBN 978-94-011-1078-5 (ebook) Printed on acid-free paper AII Rights Reserved 1994 Springer Science+Business Media Oordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1 st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

in honour of Asim Orhan Barut

CONTENTS Preface... xi 1. E. A. Tanner and R. Wilson... 1 Noncompact Lie groups, their algebras and some of their applications. Lie Groups and Lie Algebras 2. S. Helgason... 55 Harish-Chandra's c-function. A mathematical jewel. 3. E. Van Den Ban, M. Flensted-Jensen and H. Schlichtkrull... 69 Basic harmonic analysis on pseudo-riemannian symmetric spaces. 4. A. O. Barut... 103 The extensions of space-time. Physics in the 8-dimensional homogeneous space 1) = SU(2,2)/K. 5. P. Budinich... 123 Ordinary - and momentum - space conformal compactifications: Some possible observable consequences. 6. J. Hilgert... 141 Radon transform on half planes via group theory. 7. B. Speh... 147 Analytic torsion and automorphic forms. 8. A. Figa-Talamanca... 157 Diffusion on compact ultra metric spaces. 9. J.-P. Antoine... 169 Generalized square integrability and coherent states. 10. P. Winternitz, M. A. del Olmo and M. A. Rodriguez... 181 Maximal abelian subgroups of SU(p, q) and integrable Hamiltonian systems. 11. A. Inomata and G. Junker... 199 Path integrals and Lie groups. 12. T. Hirai... 225 Representations of diffeomorphism groups and the infinite symmetric group.

viii 13. M. Anoussis... 239 Characters of Lie groups. 14. W. Schmid and K. Vilonen 243 Weyl group actions on Lagrangian cycles and Rossmann's formula. 15. E. Angelopoulos... 251 Taylor formula, tensor products, and unitarizability. 16. G. W. Mackey... 265 A connection between Lie algebra roots and weights and the Fock space construction. 17. D. J. Rowe... 285 Applications of Sp(3, R) in nuclear physics. 18. W. H. Klink... 301 Nilpotent groups and anharmonic oscillators. 19. C. H. Conley... 315 Extensions of the mass 0 helicity 0 representation of the Poincare group. 20. W. F. Heidenreich... 325 Invariant causal propagators in conformal space. 21. R. F. Streater... 333 Gauge groups, anomalies and non-abelian cohomology. 22. R. V. Moody and J. Patera... 341 The Es family of quasicrystals. 23. R. O. Wells, Jr. a.nd X. Zhou... 349 Wavelet interpolation and approximate solutions of elliptic partial differential equations. Lie Superalgebras and Lie Supergroups 24. V. Hussin and 1. M. Nieto... 367 From super Lie algebras to supergroups: Matrix realizations and the factorisation problem. 25. J. Mickelsson... 373 Current algebras as Hilbert space operator cocycles. 26. J. Niederle... 391 Non-linear realization technique - The most convenient way of deriving N = 1 supergravity.

27. L. O'Raifeartaigh... 405 Toda systems as constrained linear systems. ix Quantum Groups 28. A. Guichardet... 413 On the definitions of the quantum group 'ft'h( sl(2, k)) and the restricted dual of 'ft'h( sl( n, k)). 29. C. Fr nsdal... 423 Universal T-matrix for twisted quantum gl(n). 30. W. Pusz and S. L. Woronowicz... 453 Unitary representations of quantum Lorentz group. 31. R. Giachetti... 473 Contraction of quantum groups and lattice physics. 32. L. C. Biedenharn and M. Tarlini... 487 A quantum Poincare group and the Dirac-Coulomb problem.

PREFACE During the past two decades representations of noncompact Lie groups and Lie algebras have been studied extensively, and their application to other branches of mathematics and to physical sciences has increased enormously. Several theorems which were proved in the abstract now carry definite mathematical and physical significance. Several physical observations which were not understood before are now explained in terms of models based on new group-theoretical structures such as dynamical groups and Lie supergroups. The workshop was designed to bring together those mathematicians and mathematical physicists who are actively working in this broad spectrum of research and to provide them with the opportunity to present their recent results and to discuss the challenges facing them in the many problems that remain. The objective of the workshop was indeed well achieved. This book contains 31 lectures presented by invited participants attending the NATO Advanced Research Workshop held in San Antonio, Texas, during the week of January 3-8, 1993. The introductory article by the editors provides a brief review of the concepts underlying these lectures (cited by author [*]) and mentions some of their applications. The articles in the book are grouped under the following general headings: Lie groups and Lie algebras, Lie superalgebras and Lie supergroups, and Quantum groups, and are arranged in the order in which they are cited in the introductory article. We are very thankful to Dr. Luigi Sertorio, the Programme Director of the NATO Advanced Research Workshop, for sponsoring the meeting and for his invaluable advice at each stage of its organization. The workshop was also partially supported by The University of Texas at San Antonio, Technical Concepts Corporation, and the St. Anthony Hotel. We should like to thank, respectively, Dr. Shair Ahmad, Mr. Charles Becker and Mrs. Michele Lock for their efforts in arranging financial support. We are also grateful to Drs. Asim Barut, Alain Guichardet, and Alessandro Figit Talamanca for serving on the International Advisory Committee and giving us valuable advice. The St. Anthony Hotel, host for the worhshop, provided the participants with an outstanding academic environment accompanied by a very warm hospitality. We would like to thank each and every staff member of the hotel for their help during this workshop. Finally, especially, we thank all the contributors and participants for making the workshop a lively and fruitful meeting and the book a very useful contribution to the field of noncompact Lie groups and Lie algebras. We are very proud to publish this book in honour of Professor Asim Barut who, in fact, first recognized the need for this meeting. Elizabeth Tanner Raj Wilson San Antonio, 25 December 1993 Xl